2 Answers
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Take everything I say with a grain of salt (pun intended) as I am writing from Texas where we do nobler things with brisket. A true corned-beef-expert would be of Irish or Jewish descent, and from New York.

The meat should have relatively tough individual fibers that separate easily because of the long, wet cooking. It is pretty difficult to overcook brisket as it is so tough. Like Chuck Norris but tastier. Three hours would definitely not be enough time.

It can be fatty and that will be pretty much up to how much of the fat was trimmed off when it was butchered. An untrimmed brisket has a solid layer of fat that can be as much as a half inch thick. Obviously trimmed briskets cost more. The fat can be nice for smoking but the corned beef I have seen is not as fatty.

Fatty and mushy are not the same thing, of course. If the meat is actually mushy then I suspect that it was packed with a meat tenderizer as well as the corning salt and seasonings. If this is the case then there isn't much you could have done differently. Just don't buy that brand next time. Better yet, next time, corn it yourself.